I think this partially answers your question: One thing people try to do to "fix" the incompleteness is to just add the Godel sentence as an axiom of the mathematical system. However, each time you do this, you can create yet another new Godel sentence for the new system with the new axiom. Since it's a diagonalization argument, no amount of adding of new axioms can permanently "fix" the system and make it complete -- there will always be new Godel sentence that is true but not provable.